Problems to Ponder (November edition)

Welcome to the November edition of Problems to Ponder! This month’s problems have been curated by Michael Pruner, president of the British Columbia Association of Mathematics Teachers (BCAMT). The tasks are released on a weekly basis through the BCAMT listserv, and are also shared via Twitter (@BCAMT) and on the BCAMT website. This post features only a subset of the problems shared by Michael last month – head to the BCAMT website for the full set!

I am calling these problems ‘competency tasks’ because they seem to fit quite nicely with the curricular competencies in the British Columbia revised curriculum. They are non-content based, so that all students should be able to get started and investigate by drawing pictures, making guesses, or asking questions. When possible, extensions are provided so that you can keep your students in flow during the activity. Although they may not fit under a specific topic for your course, the richness of the mathematics comes out when students explain their thinking or show creativity in their solution strategies.

I think it would be fun and more valuable for everyone if we shared our experiences with the tasks. Take pictures of students’ work and share how the tasks worked with your class through the BCAMT listserv [which currently connects nearly one thousand educators from across the province, country, and even the world! –Ed.] so that others may learn from your experiences.

Intermediate and Secondary Tasks (Grades 4-12)

October 2, 2016

Rope Around the EarthA rope is wrapped tight around the Earth along the equator. The rope is cut, and 1 m of rope is added to the length and then stitched back together. A big green super hero lifts the rope and throws it so hard that it enters into circular orbit the Earth (bonus for the physicists: How fast does the rope need to spin to be in orbit?). How high is the rope hovering over planet Earth? (Earth’s radius is 6371 km.)

Extensions: What about Jupiter, the Sun, or a basketball? Instead of spinning the rope, the super hero lifts the rope straight up until it is tight again. How high can she lift the rope?

October 9, 2016

The Chess BoardHow many squares are there on a chess board? And no, the answer is not 64.

Extensions: How many rectangles? How many triangles in the figure below?

October 16, 2016

Number PatternConsider the following pattern of 5 whole numbers, where each number is the sum of the previous two numbers: 3, 12, 15, 27, 42. I want the 5th number to be 100. Find all the whole “seed” numbers that will make this so (3 and 12 are the seed numbers in the above sequence).

Source: Peter Liljedahl

October 23, 2016

Cartesian ChaseThis is a game for two players on a rectangular grid with a fixed number of rows and columns. Play begins in the bottom-left-hand square, where the first player puts his mark. On his turn, a player may put his mark into a square directly above, directly to the right of, or diagonally above and to the right of the last mark made by his opponent. Play continues in this fashion, and the winner is the player who gets their mark in the upper-right-hand corner first. Find a way of winning that your great aunt Maud could understand and use.

Extensions: What if you cannot move diagonally? What if the top right square means that the player loses?

Silver CoinsYou have 10 silver coins in your pocket (silver means that the coins could be any of nickels, dimes, or quarters). How many different amounts of money could you have?

Primary Tasks (Grades K-3)

October 9, 2016

Materials

10 or more snap cubes per student

This is an activity that children can work on in groups. Each child makes a train of connecting cubes of a specified number. On the signal “Snap,” children break their trains into two parts and hold one hand behind their back.

Children take turns going around the circle showing their remaining cubes. The other children work out the full number combination.

Take your class outside and have students collect 5 of an object (leaves, rocks, etc..). The task is for students to work in groups and find different ways to make 5. How many ways can you make 5? How can you show all of your ways?

Extensions: How about 4? How about 6?

October 23, 2016

Give students time to explore the attributes of various 3-D shapes. Have them identify the faces, edges and vertices of the 3-D shapes. Present various problems for them to solve:

If you had 3 cones, 2 cylinders, and a sphere, how many faces would you have? How do you know?

You have 1 cube and your friend has 4 cylinders. Who has more faces? How do you know?

I have some objects and in total I counted 8 faces. What might the objects be? Explain your thinking.

I have a collection of objects that have 7 faces and a point. What shapes could they be? Explain your thinking.

I used digit cards to create a 2-digit number pattern. The wind blew the cards and mixed them up. How might you place the loose digit cards into the following to complete a pattern? How do you know? How might you extend the pattern?

Michael Pruner is the current president of the British Columbia Association of Mathematics Teachers (BCAMT) and a full-time mathematics teacher at Windsor Secondary School in North Vancouver. He teaches using the Thinking Classroom model where students work collaboratively on tasks to develop both their mathematical competencies and their understanding of the course content.